Example 7
Find the values of x for which the function f (x)
= x^{3}  12x + 5 is (I) decreasing and (II) increasing.
Solution : f (x) =
x^{3}  12x + 5
\
f ' (x) = 3x^{2}  12
= 3 (x^{2}
 4)
Now ' f ' is decreasing
if f ' (x) < 0
i.e. 3 (x^{2}
 4) <0
i.e. (x^{2}
 4) < 0
i.e. x^{2}
< 4
i.e. 2 < x <
2
\ f
(x) decreases in (2, 2)
Similarly 'f ' increases
if f ' ( x ) > 0
i.e. 3(x^{2}
 4 ) > 0
i.e. x^{2}
> 0
i.e. x < 2 and
x >2
Example 8
Show that f (x) = x^{3}  6x^{2}
+ 15x + 7 is always increasing.
Solution : f (x) = x^{3}
 6x^{2} + 15x + 7
\ f ' (x) = 3x^{2}
 12x + 15
= 3 (x^{2}  4x
+5 )
\ f ' (x)
= 3 [ x^{2}  4x + 4 + 1]
= 3 [ (x 2)^{2}
+ 1 ]
Now 3 [ (x  2)^{2} + 1] > 0
for all x
\ f ' (x)
> 0 for all x
\ f (x) is
always increasing
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